The first thing to understand is exactly what causes the sound we hear when our brakes makes that terrible sound. The reason for it is
an instability due to the friction forces leading to self-excited vibrations. That is, there is, for some reason, an non-constant amount of braking friction (between the pad and the rotor). This creates vibrations that then are audible. It's similar to the sound you can create when you wet your finger, and run it along the top of a wine glass. When the vibrations match the natural frequency of the wine glass, the feed on themselves and give off a tone. More or less the same with brakes.

So, the trick is to find a way to clean the surface of the rotor and/or the face of the pads so that the friction that is created when you apply the brakes is CONSTANT....thus, no vibrations. 100% of the problem is in the pads and/or the rotor. To change the entire brake system would be a waste.

So....clean the rotor, BOTH sides, with a non-oily solvent such as acetone or alcohol. Or you could use a non-polar solvent like MEK, mineral spirits, xylene (goof off) etc....BUT make sure to then remove any and all residue that may have come from the solvent. To do that, I use acetone or isopropyl alcohol. If that doesn't solve the problem then I'd look to the pads; but, you already said your LBS changed the pads.

One other thing to keep in mind is that all brakes will squeal at some time, under some conditions. It doesn't mean they're bad....it just means there is an uneven friction creating a vibration. Weather conditions often play a roll. One day my brakes are fine. The next, they squeal when they're cold. The next they may squeal only when hot.